Large $z$ Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane

Abstract

In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as $n\times n$ Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as $n$-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane.

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• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
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Cited by in corpus (3)

• Open Problems for Painlevé Equations
• Generalised Airy Polynomials
• Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters